A008278 → x ↦ x+1 (shift +1) → θ+1

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No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:      1;
  1:      2,       2;
  2:      5,      10,       3;
  3:     15,      46,      30,       4;
  4:     52,     218,     228,      76,       5;
  5:    203,    1088,    1593,     896,     180,       6;
  6:    877,    5752,   10923,    8816,    3155,     414,      7;
  7:   4140,   32226,   75624,   80356,   42355,   10494,    938,      8;
  8:  21147,  190990,  535692,  710524,  500855,  187986,  33866,   2104,    9;
  9: 115975, 1194310, 3907353, 6239152, 5547365, 2830962, 796649, 107632, 4680, 10;

The row polynomials satisfy -(x-n*x^3)*P(n,x) = (2*n+(n^2+5*n-3)*x-4*x^2-(n^3+5*n^2+3*n-3)*x^3-(2*n^3+4*n^2-2*n-4)*x^4-(n^3-n)*x^5)*P(n-2,x) - (1+(n+4)*x+(2*n+4)*x^2-(n^2+2*n+2)*x^3-(3*n^2+7*n+5)*x^4-(3*n^2+5*n+2)*x^5-(n^2+n)*x^6)*d/dx P(n-2,x) - (2+(2*n+3)*x-2*x^2-(2*n^2+4*n+2)*x^3-2*n^2*x^4)*P(n-1,x) + (x+2*x^2-(n-1)*x^3-2*n*x^4-n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + (n^2 + 5*n - 3)*x - 4*x^2 - (n^3 + 5*n^2 + 3*n - 3)*x^3 - (2*n^3 + 4*n^2 - 2*n - 4)*x^4 - (n^3 - n)*x^5)*P[n - 2, x] - (1 + (n + 4)*x + (2*n + 4)*x^2 - (n^2 + 2*n + 2)*x^3 - (3*n^2 + 7*n + 5)*x^4 - (3*n^2 + 5*n + 2)*x^5 - (n^2 + n)*x^6)*D[P[n - 2, x], x] - (2 + (2*n + 3)*x - 2*x^2 - (2*n^2 + 4*n + 2)*x^3 - 2*n^2*x^4)*P[n - 1, x] + (x + 2*x^2 - (n - 1)*x^3 - 2*n*x^4 - n*x^5)*D[P[n - 1, x], x])/(-x + n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + (n^2 + 5*n - 3)*x - 4*x^2 - (n^3 + 5*n^2 + 3*n - 3)*x^3 - (2*n^3 + 4*n^2 - 2*n - 4)*x^4 - (n^3 - n)*x^5)*v[i - 2] - (1 + (n + 4)*x + (2*n + 4)*x^2 - (n^2 + 2*n + 2)*x^3 - (3*n^2 + 7*n + 5)*x^4 - (3*n^2 + 5*n + 2)*x^5 - (n^2 + n)*x^6)*deriv(v[i - 2], x) - (2 + (2*n + 3)*x - 2*x^2 - (2*n^2 + 4*n + 2)*x^3 - 2*n^2*x^4)*v[i - 1] + (x + 2*x^2 - (n - 1)*x^3 - 2*n*x^4 - n*x^5)*deriv(v[i - 1], x))/(-x + n*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-9b9aa33d8b05d5b3
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-9b9aa33d8b05d5b3
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-9b9aa33d8b05d5b3

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image euler_polynomial, shift 0 e-9b9aa33d8b05d5b3

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image affine, shift 1 e-236d0fb15dc6eca0

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-8e580efc86a38a12 root: theorem check

          Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • preserver_image reversal, shift 0 e-3256729839b26765

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-995393ce4e49a8d6 root: theorem check

              Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

            • basis_map A008277 e-c52960bc5e97caa0

              Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

              • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • parent_provers periodic_product e-6e85c0971e84e272 root: theorem check

                  Parent-row provers of polytool polytool ?

                • preserver_image mobius, shift 0 e-ad67990681a86223

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
                    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                    • ↵ parent_provers periodic_product e-6e85c0971e84e272 (expanded above)
            • preserver_image reversal, shift 0 e-d527982dc7c0aa3b

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-8e580efc86a38a12 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.